118
Introducing (C.3) into (C.2) yields
gvi gh20
L = ^ + + + fdz+^-(h0 + t¡) +
eVKoVhi [ f dz + eVhi { Vdz + [ fMdz\ +
J-h0 U-h J-Kc )
(yoif, |y +0(t'
2
The Lagrangian L is assumed to be of the form
(C.4)
It then follows that
L Lq + fLi + e2Z2 + f? L$ + 1
o d!-iU £££(*.+,)
and
(C.5)
(C.6)
1 = gf>m + i, + v^oV^! { r v*/ dz + r fzdz) + m (V^o)2 (c.7)
0 fl U-fco /-* J 2
f [fVhf + / /*] + Vi^hoifx |#J (c.8)
* fto
To obtain the desired linear governing equation is varied with respect to i and then
is varied with respect to r¡\ in order to eliminate r¡\. The variational principle holds that the
Lagrangian is constant under small variations of the unknown parameters. In mathematical
terms
and
DL
D4>!
DL
Dr)!
= 0
= 0
(C.9)
(cio)
Since the ordering parameter is arbitrary the above holds for all L, = 0,1,2, . In the
2 term i is expressed in terms of its derivatives. This requires integration by parts to